# Strilets O. V.

### On simple $n$-tuples of subspaces of a Hilbert space

Samoilenko Yu. S., Strilets O. V.

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1668-1703

This survey is devoted to the structure of “simple” systems $S = (H;H_1,…,H_n)$ of subspaces $H_i,\; i = 1,…, n,$ of a Hilbert space $H$, i.e., $n$-tuples of subspaces such that, for each pair of subspaces $H_i$ and $H_j$, the angle $0 < θ_{ij} ≤ π/2$ between them is fixed. We give a description of “simple” systems of subspaces in the case where the labeled graphs naturally associated with these systems are trees or unicyclic graphs and also in the case where all subspaces are one-dimensional. If the cyclic range of a graph is greater than or equal to two, then the problem of description of all systems of this type up to unitary equivalence is *-wild.

### On the *-representation of one class of algebras associated with Coxeter graphs

Popova N. D., Samoilenko Yu. S., Strilets O. V.

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 545–556

We investigate *-representations of a class of algebras that are quotient algebras of the Hecke algebras associated with Coxeter graphs. A description of all unitarily nonequivalent irreducible *-representations of finite-dimensional algebras is given. We prove that only trees that have at most one edge of type *s* > 3 define algebras of finite Hilbert type for all values of parameters.

### On the growth of deformations of algebras associated with Coxeter graphs

Popova N. D., Samoilenko Yu. S., Strilets O. V.

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 826–837

We investigate a class of algebras that are deformations of quotient algebras of group algebras of Coxeter groups. For algebras from this class, a linear basis is found by using the “diamond lemma.” A description of all finite-dimensional algebras of this class is given, and the growth of infinite-dimensional algebras is determined.

### On the identities in algebras generated by linearly connected idempotents

Rabanovych V. I., Samoilenko Yu. S., Strilets O. V.

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 782–795

We investigate the problem of the existence of polynomial identities (PI) in algebras generated by idempotents whose linear combination is equal to identity. In the case where the number of idempotents is greater than or equal to five, we prove that these algebras are not *PI*-algebras. In the case of four idempotents, in order that an algebra be a *PI*-algebra, it is necessary and sufficient that the sum of the coefficients of the linear combination be equal to two. In this case, these algebras are *F* _{4}-algebras.

### On Identities in Algebras $Q_{n,λ}$ Generated by Idempotents

Rabanovych V. I., Samoilenko Yu. S., Strilets O. V.

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1380-1390

We investigate the presence of polynomial identities in the algebras $Q_{n,λ}$ generated by $n$ idempotents with the sum $λe$ ($λ ∈ C$ and $e$ is the identity of an algebra). We prove that $Q_{4,2}$ is an algebra with the standard polynomial identity $F_4$, whereas the algebras $Q_{4,2},\; λ ≠ 2$, and $Q_{n,λ},\; n ≥ 5$, do not have polynomial identities.